Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids

Abstract

Solving the dynamics of equilibrium dense liquids is a notoriously difficult problem [1]. In the low-density limit, or at short times, the solution is obtained via kinetic theory, while at any density, hydrodynamics provides a description of the dynamics at large length and time scales. However, at high densities or low temperatures, kinetic theory breaks down, and the hydrodynamic regime is pushed to scales that are much larger than any experimentally relevant scale. In this strongly interacting "supercooled liquid" regime, dynamics is slow, viscosity is high, and both correlation functions and transport coefficients display non-trivial behavior that is neither captured by kinetic theory nor by hydrodynamics. The only microscopic description of this regime is obtained by the Mode-Coupling Theory (MCT) [2,3], which is a set of closed equations derived from a series of poorly controlled approximations of the true dynamics. Despite its non-systematic nature [4][5][6][7], MCT accurately describes the initial slowing down of liquid dynamics upon supercooling, including the wavevector dependence of correlation functions [8].Interestingly, in the formal limit in which the spatial dimension d goes to infinity, liquid thermodynamics reduces to the calculation of the second virial coefficient [9][10][11][12]. Based on this observation, a first attempt to solve exactly liquid dynamics for d → ∞ was presented in [13] (see also [14]), but the full dynamical mean field theory (DMFT) that describes exactly the equilibrium dynamics in d → ∞ was only derived recently, via a second-order virial expansion on trajectories [15] or via a dynamic cavity method [16,17]. The DMFT provides a set of closed one-dimensional integro-differential equations, which exactly describe the many-body liquid dynamics in the thermodynamic limit, and are similar in structure to those obtained for quantum systems in the same limit [18]. We refer the reader to [17,[19][20][21] for a detailed review of the solution of liquid dynamics in infinite dimensions, including its extension to the out-of-equilibrium setting.Unfortunately, the analytical solution of the DMFT equations is out of reach. In this work, we present their numerical solution, obtained through an iterative method and a straightforward discretization of time. This strategy, however, only works for differentiable interaction potentials. We thus discuss how to derive the DMFT of hard spheres via a non-trivial limit of a soft sphere interaction. We present numerical results for soft and hard spheres, supported by analytical computations at low densities, in the short time limit, and at long times in the glass phase.The paper is organized as follows. In section II we review the basic equations of the DMFT of liquids. In section III, we discuss the Hard Sphere limit of DMFT. In section IV, we discuss the discretization and convergence algorithms used in this work. In section V, we present the numerical solution for Soft and Hard Spheres. Finally, in section VI we draw our conclusions and present s...

Gradient descent dynamics and the jamming transition in infinite dimensions

“…Even if the available computational resources do not allow us to reach convergence in the cutoff at high density, we can still elucidate a somehow counterintuitive behavior of the dynamics, i.e. the increase of the MSD with density at fixed d. The situation is indeed inverted with respect to systems in thermal equilibrium, where a density increase yields the transition from diffusive to arrested dynamics and the asymptotic MSD decreases upon increasing density [39]; in this athermal case, in the low-density unjammed phase the tracer reaches the boundary of a zero-energy lake, and stops its motion there, leading to a finite displacement from the origin. Upon increasing the density, the tracer starts its dynamics from a higher energy level and it surfes over several energy valleys before reaching the zero-energy region.…”

“…the discussion in [38, chapter 9]. Unfortunately, these equations are particularly difficult to solve numerically [39]. We discuss here some analytical results, while the numerical solution will be discussed in section III.…”

“…Because the numerical solution of the DMFT equations is difficult, it is useful to consider here a low-density, dilute limit, to have an idea of what to expect at higher densities. In the very low-density limit, we can neglect all the kernels, because they have a factor ϕ in front [39]. The equations for correlation and response then reduce to…”

“…The outcome of the first iteration is then given by the dilute solution at ϕ 1 derived in section II D, as we explicitly checked in the numerical implementation. This algorithm is the extension to the out-of-equilibrium case of the equilibrium algorithm implemented in [39].…”

Tagged-particle dynamics in confined colloidal liquids

“…To systematically investigate colloidal liquids, several first-principle theories have been developed, including mode-coupling theory (MCT) [42][43][44], self-consistent generalized Langevin dynamics [45] and dynamical meanfield theory [46,47]. For supercooled bulk liquids these theories successfully predict several important features like the slowing down of transport, stretching of the intermediate scattering function, as well as a two-step powerlaw relaxation behavior [48][49][50].…”